\documentclass[dvipsnames,14pt,t]{beamer}\usepackage{beamerthemeplainculight}\usepackage[T1]{fontenc}\usepackage[latin1]{inputenc}\usepackage{mathpartir}\usepackage[absolute,overlay]{textpos}\usepackage{ifthen}\usepackage{tikz}\usepackage{pgf}\usepackage{calc} \usepackage{ulem}\usepackage{courier}\usepackage{listings}\renewcommand{\uline}[1]{#1}\usetikzlibrary{arrows}\usetikzlibrary{automata}\usetikzlibrary{shapes}\usetikzlibrary{shadows}\usetikzlibrary{positioning}\usetikzlibrary{calc}\usepackage{graphicx} \definecolor{javared}{rgb}{0.6,0,0} % for strings\definecolor{javagreen}{rgb}{0.25,0.5,0.35} % comments\definecolor{javapurple}{rgb}{0.5,0,0.35} % keywords\definecolor{javadocblue}{rgb}{0.25,0.35,0.75} % javadoc\lstset{language=Java, basicstyle=\ttfamily, keywordstyle=\color{javapurple}\bfseries, stringstyle=\color{javagreen}, commentstyle=\color{javagreen}, morecomment=[s][\color{javadocblue}]{/**}{*/}, numbers=left, numberstyle=\tiny\color{black}, stepnumber=1, numbersep=10pt, tabsize=2, showspaces=false, showstringspaces=false}\lstdefinelanguage{scala}{ morekeywords={abstract,case,catch,class,def,% do,else,extends,false,final,finally,% for,if,implicit,import,match,mixin,% new,null,object,override,package,% private,protected,requires,return,sealed,% super,this,throw,trait,true,try,% type,val,var,while,with,yield}, otherkeywords={=>,<-,<\%,<:,>:,\#,@}, sensitive=true, morecomment=[l]{//}, morecomment=[n]{/*}{*/}, morestring=[b]", morestring=[b]', morestring=[b]"""}\lstset{language=Scala, basicstyle=\ttfamily, keywordstyle=\color{javapurple}\bfseries, stringstyle=\color{javagreen}, commentstyle=\color{javagreen}, morecomment=[s][\color{javadocblue}]{/**}{*/}, numbers=left, numberstyle=\tiny\color{black}, stepnumber=1, numbersep=10pt, tabsize=2, showspaces=false, showstringspaces=false}% beamer stuff \renewcommand{\slidecaption}{AFL 03, King's College London, 10.~October 2012}\newcommand{\bl}[1]{\textcolor{blue}{#1}} \newcommand{\dn}{\stackrel{\mbox{\scriptsize def}}{=}}% for definitions\begin{document}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}<1>[t]\frametitle{% \begin{tabular}{@ {}c@ {}} \\[-3mm] \LARGE Automata and \\[-2mm] \LARGE Formal Languages (3)\\[3mm] \end{tabular}} %\begin{center} %\includegraphics[scale=0.3]{pics/ante1.jpg}\hspace{5mm} %\includegraphics[scale=0.31]{pics/ante2.jpg}\\ %\footnotesize\textcolor{gray}{Antikythera automaton, 100 BC (Archimedes?)} %\end{center}\normalsize \begin{center} \begin{tabular}{ll} Email: & christian.urban at kcl.ac.uk\\ Of$\!$fice: & S1.27 (1st floor Strand Building)\\ Slides: & KEATS (also home work is there)\\ & \alert{\bf (I have put a temporary link in there.)}\\ \end{tabular} \end{center}\end{frame}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Last Week\end{tabular}}Last week I showed you\begin{itemize}\item one simple-minded regular expression matcher (which however does not work in all cases), and\bigskip\item one which works provably in all cases\begin{center}\bl{matcher r s} \;\;if and only if \;\; \bl{s $\in$ $L$(r)}\end{center} \end{itemize}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}The Derivative of a Rexp\end{tabular}}\begin{center}\begin{tabular}{@ {}l@ {\hspace{2mm}}c@ {\hspace{2mm}}l@ {\hspace{-10mm}}l@ {}} \bl{der c ($\varnothing$)} & \bl{$\dn$} & \bl{$\varnothing$} & \\ \bl{der c ($\epsilon$)} & \bl{$\dn$} & \bl{$\varnothing$} & \\ \bl{der c (d)} & \bl{$\dn$} & \bl{if c $=$ d then $\epsilon$ else $\varnothing$} & \\ \bl{der c (r$_1$ + r$_2$)} & \bl{$\dn$} & \bl{(der c r$_1$) + (der c r$_2$)} & \\ \bl{der c (r$_1$ $\cdot$ r$_2$)} & \bl{$\dn$} & \bl{if nullable r$_1$}\\ & & \bl{then ((der c r$_1$) $\cdot$ r$_2$) + (der c r$_2$)}\\ & & \bl{else (der c r$_1$) $\cdot$ r$_2$}\\ \bl{der c (r$^*$)} & \bl{$\dn$} & \bl{(der c r) $\cdot$ (r$^*$)}\\ \end{tabular}\end{center}``the regular expression after \bl{c} has been recognised'' \end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]For this we defined the set \bl{Der c A} as\begin{center}\bl{Der c A $\dn$ $\{$ s $|$ c::s $\in$ A$\}$ } \end{center}which is called the semantic derivative of a setand proved \begin{center}\bl{$L$(der c r) $=$ Der c ($L$(r))}\end{center}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}The Idea of the Algorithm\end{tabular}}If we want to recognise the string \bl{abc} with regular expression \bl{r}then\medskip\begin{enumerate}\item \bl{Der a ($L$(r))}\pause\item \bl{Der b (Der a ($L$(r)))}\item \bl{Der c (Der b (Der a ($L$(r))))}\pause\item finally we test whether the empty string is in set\pause\medskip\end{enumerate}The matching algorithm works similarly, just over regular expression than sets.\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]Input: string \bl{abc} and regular expression \bl{r} \begin{enumerate}\item \bl{der a r}\item \bl{der b (der a r)}\item \bl{der c (der b (der a r))}\pause\item finally check whether the latter regular expression can match the empty string\end{enumerate}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]We need to prove\begin{center}\bl{$L$(der c r) $=$ Der c ($L$(r))}\end{center}by induction on the regular expression.\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Proofs about Rexp\end{tabular}}\begin{itemize}\item \bl{$P$} holds for \bl{$\varnothing$}, \bl{$\epsilon$} and \bl{c}\bigskip\item \bl{$P$} holds for \bl{r$_1$ + r$_2$} under the assumption that \bl{$P$} alreadyholds for \bl{r$_1$} and \bl{r$_2$}.\bigskip\item \bl{$P$} holds for \bl{r$_1$ $\cdot$ r$_2$} under the assumption that \bl{$P$} alreadyholds for \bl{r$_1$} and \bl{r$_2$}.\item \bl{$P$} holds for \bl{r$^*$} under the assumption that \bl{$P$} alreadyholds for \bl{r}.\end{itemize}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Proofs about Natural Numbers\\ and Strings\end{tabular}}\begin{itemize}\item \bl{$P$} holds for \bl{$0$} and\item \bl{$P$} holds for \bl{$n + 1$} under the assumption that \bl{$P$} alreadyholds for \bl{$n$}\end{itemize}\bigskip\begin{itemize}\item \bl{$P$} holds for \bl{\texttt{""}} and\item \bl{$P$} holds for \bl{$c\!::\!s$} under the assumption that \bl{$P$} alreadyholds for \bl{$s$}\end{itemize}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[t]\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}\begin{center} \begin{tabular}{@ {}rrl@ {\hspace{13mm}}l} \bl{r} & \bl{$::=$} & \bl{$\varnothing$} & null\\ & \bl{$\mid$} & \bl{$\epsilon$} & empty string / "" / []\\ & \bl{$\mid$} & \bl{c} & character\\ & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$} & sequence\\ & \bl{$\mid$} & \bl{r$_1$ + r$_2$} & alternative / choice\\ & \bl{$\mid$} & \bl{r$^*$} & star (zero or more)\\ \end{tabular}\bigskip\pause \end{center}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Languages\end{tabular}}A \alert{language} is a set of strings.\bigskipA \alert{regular expression} specifies a set of strings or language.\bigskipA language is \alert{regular} iff there existsa regular expression that recognises all its strings.\bigskip\bigskip\pause\textcolor{gray}{not all languages are regular, e.g.~\bl{a$^n$b$^n$}.}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[t]\frametitle{\begin{tabular}{c}Regular Expressions\end{tabular}}\begin{center} \begin{tabular}{@ {}rrl@ {\hspace{13mm}}l} \bl{r} & \bl{$::=$} & \bl{$\varnothing$} & null\\ & \bl{$\mid$} & \bl{$\epsilon$} & empty string / "" / []\\ & \bl{$\mid$} & \bl{c} & character\\ & \bl{$\mid$} & \bl{r$_1$ $\cdot$ r$_2$} & sequence\\ & \bl{$\mid$} & \bl{r$_1$ + r$_2$} & alternative / choice\\ & \bl{$\mid$} & \bl{r$^*$} & star (zero or more)\\ \end{tabular}\bigskip \end{center}How about ranges \bl{[a-z]}, \bl{r$^\text{+}$} and \bl{!r}?\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Negation of Regular Expr's\end{tabular}}\begin{itemize}\item \bl{!r} \hspace{6mm} (everything that \bl{r} cannot recognise)\medskip\item \bl{$L$(!r) $\dn$ UNIV - $L$(r)}\medskip\item \bl{nullable (!r) $\dn$ not (nullable(r))}\medskip\item \bl{der\,c\,(!r) $\dn$ !(der\,c\,r)}\end{itemize}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Regular Exp's for Lexing\end{tabular}}Lexing separates strings into ``words'' / components.\begin{itemize}\item Identifiers (non-empty strings of letters or digits, starting with a letter)\item Numbers (non-empty sequences of digits omitting leading zeros)\item Keywords (else, if, while, \ldots)\item White space (a non-empty sequence of blanks, newlines and tabs)\item Comments\end{itemize}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\mode<presentation>{\begin{frame}[c]\frametitle{\begin{tabular}{c}Automata\end{tabular}}A deterministic finite automaton consists of:\begin{itemize}\item a set of states\item one of these states is the start state\item some states are accepting states, and\item there is transition function\medskip \smallwhich takes a state as argument and a character and produces a new state\smallskip\\this function might not always be defined\end{itemize}\end{frame}}%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}%%% Local Variables: %%% mode: latex%%% TeX-master: t%%% End: